Boundary graphs: The limit case of a spectral property

Abstract:A graph is walk-regular if the number of closed walks of length rooted at a given vertex is a constant through all the vertices for all . For a walk-regular graph G with d+1 different eigenvalues and spectrally maximum diameter D = d, we study the geometry of its d-spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three-dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their t-spreads or vertices at distance t from each other. ᭧

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“…So, the points in P h (U) are the vertices of a simplex with center c r , radius R and side L as claimed in (9).…”

Section: Proof Computing Wmentioning

confidence: 94%

“…Fiol et al [9] defined the crossed (uv−)local multiplicities of h , denoted by m uv ( h ), as the uv-entries of the idempotents (u = v). Now in terms of the orthogonal projection of the canonical vectors e u , the crossed local multiplicities are obtained by the Euclidean scalar products…”

Section: The Crossed Local Multiplicities and The Cosinesmentioning

confidence: 99%

The geometry oft‐spreads ink‐walk‐regular graphs

Dalfó

Fiol

Garriga

2010

Journal of Graph Theory

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“…By analogy with the local multiplicities, which correspond to the diagonal entries of the idempotents, Fiol, Garriga, and Yebra [13] defined the crossed (uv-)local multiplicities of eigenvalue λ i , denoted by m uv (λ i ), as…”

Section: Graphs and Their Spectramentioning

confidence: 99%

Edge-distance-regular graphs

Cámara

Dalfó

Fíbrega

et al. 2011

Journal of Combinatorial Theory, Series A

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Edge-distance-regularity is a concept recently introduced by the authors which is similar to that of distance-regularity, but now the graph is seen from each of its edges instead of from its vertices. More precisely, a graph Γ with adjacency matrix A is edge-distance-regular when it is distance-regular around each of its edges and with the same intersection numbers for any edge taken as a root. In this paper we study this concept, give some of its properties, such as the regularity of Γ , and derive some characterizations. In particular, it is shown that a graph is edge-distance-regular if and only if its k-incidence matrix is a polynomial of degree k in A multiplied by the (standard) incidence matrix. Also, the analogue of the spectral excess theorem for distance-regular graphs is proved, so giving a quasi-spectral characterization of edge-distance-regularity. Finally, it is shown that every nonbipartite graph which is both distance-regular and edge-distance-regular is a generalized odd graph.

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“…Given two vertices i, j and any eigenvalue λ l , Garriga, Yebra and the first author in [16], introduced the concept of crossed (ij-)local multiplicity of λ l as m ij (λ l ) := z il , z jl . Note that this corresponds to the ij-entry of the idempotent E l since, using the symmetric character of E l and property (a.1),…”

Section: The Local Multiplicitymentioning

confidence: 99%

“…More precisely, ecc(i) [16].) From the i-local spectrum (5), it is natural to consider the analogous function of the characteristic polynomial, which we call the i-local characteristic function, defined as:…”

Section: The Local Spectrummentioning

confidence: 99%

The Local Spectra of Line Graphs

Fiol

Riera

2007

Electronic Notes in Discrete Mathematics

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The local spectrum of a graph G = (V, E), constituted by the standard eigenvalues of G and their local multiplicities, plays a similar role as the global spectrum when the graph is "seen" from a given vertex. Thus, for each vertex i ∈ V , the i-local multiplicities of all the eigenvalues add up to 1; whereas the multiplicity of each eigenvalue λ l ∈ ev G is the sum, extended to all vertices, of its local multiplicities.In this work, using the interpretation of an eigenvector as a charge distribution on the vertices, we compute the local spectrum of the line graph LG in terms of the local spectrum of the (regular o semiregular) graph G it derives from. Furthermore, some applications of this result are derived as, for instance, some results related to the number of cycles.

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Boundary graphs: The limit case of a spectral property

Cited by 25 publications

References 10 publications

The geometry oft‐spreads ink‐walk‐regular graphs

The geometry oft‐spreads ink‐walk‐regular graphs

Edge-distance-regular graphs

The Local Spectra of Line Graphs

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